Euler's elastica is defined as planar curves γ that minimize energy functionals of the form:
                              E          ⁡                      (            γ            )                          =                              ∫            γ                    ⁢                                    (                              a                +                                  b                  ⁢                                                                          ⁢                                      κ                    2                                                              )                        ⁢                                                  ⁢                          ⅆ              s                                                          (        1.1        )            where κ is the curvature of the curve, s is arc length, and constants a and b are both greater than zero. This equation is of importance in various applications, so numerous papers have appeared attempting to solve it [1-6]. One of contributions in the variational approach is due to Mumford, Nitzberg, and Shiota [7], where they proposed a method for segmenting an image into objects which should be ordered according to their depth in the scene. In order to connect T-junctions at the occluding boundaries of objects, they looked for a continuation curve Γ which minimizes (1.1). Following the work [7] and observing the importance of level lines for image representation, the authors in [8] adapted the variational continuation framework of [7] to level line structures. They proposed a variational formulation for the recovery of the missing parts of a grey level image. They referred to this interpolation process as disocclusion. Consider an image from which a portion (“domain”) D is “in-painted” (that is, missing from the data set). After omitting the angular terms and taking a domain {tilde over (D)} slightly bigger than the inpainting domain D such that D⊂⊂{tilde over (D)}, their minimization criterion becomes:
                    E        =                                            ∫                              -                ∞                            ∞                        ⁢                                          ∑                                  Γ                  ∈                                      F                    λ                                                              ⁢                                                          ⁢                                                ∫                  Γ                                ⁢                                  (                                      a                    +                                          b                      ⁢                                                                                                  κ                                                                          p                                                                              )                                                              |                                          ⁢                                    ⅆ                              H                1                                      ⁢                                                  ⁢                          ⅆ              λ                                                          (        1.2        )            where p≧1, H1 denotes the one-dimensional Hausdorff measure, and the elements of Fλ are the union of completion curves and the restrictions to D\{tilde over (D)} of the associated level lines. In other words, in the boundary region where {tilde over (D)} extends beyond the region D where data is missing, there are a number of intensity levels labeled by variable λ (e.g. grey scale levels). A given reconstruction of the missing data can be considered as a set of lines {Fλ} extending across the region {tilde over (D)}, with one or more lines Fλ for each value of λ. For a given value of λ, the sum over Γ corresponds to taking Γ as the lines {Fλ} successively, calculating a component of the energy for each line, and adding those components. Then an integral over λ is performed to give the equation (1.2) which defines an energy E for the entire image. The lines {Fλ} are then sought which minimize this energy.
In order to study the inpainting problem from the viewpoint of the calculus of variations, Ambrosio and Masnou [9] rewrite Eqn. (1.2), assuming that the curves F are the level lines of a smooth function u, asE(u)=∫−∞∞(∫∂(u≧λ)∩{tilde over (D)}(a+b|κ|P)d1)dλ  (1.3)
Using the change of variable, the energy functional becomes:
                              E          ⁡                      (            u            )                          =                              ∫                          D              ~                                ⁢                                    (                              a                +                                  b                  ⁢                                                                                                                                      ∇                                                      ·                                                                                          ∇                                u                                                                                                                                                              ∇                                  u                                                                                                                                                                                                                                                ⁢                                                                                                            p                                                              )                        ⁢                                                                            ∇                  u                                                            .                                                          (        1.4        )            
As noted in [9], this criterion makes sense only for a certain class of smooth functions and needs to be relaxed in order to deal with more general functions. So they extend the energy functional (1.4) to the whole L1 (2):
                              E          ⁡                      (            u            )                          =                  {                                                                                          ∫                                          D                      ~                                                        ⁢                                                            (                                              a                        +                                                  b                          ⁢                                                                                                                                                                                      ∇                                                                      ·                                                                                                                  ∇                                        u                                                                                                                                                                                                      ∇                                          u                                                                                                                                                                                                                                                                                                                        ⁢                                                                                                                                                    p                                                                                              )                                        ⁢                                                                                        ∇                        u                                                                                                                                                                                    if                    ⁢                                                                                  ⁢                    u                                    ∈                                                            C                      2                                        ⁡                                          (                                              ℝ                        2                                            )                                                                                                                          ∞                                                                                                        if                      ⁢                                                                                          ⁢                      u                                        ∈                                                                                            L                          1                                                ⁡                                                  (                                                      ℝ                            2                                                    )                                                                    ⁢                      \                      ⁢                                                                        C                          2                                                ⁡                                                  (                                                      ℝ                            2                                                    )                                                                                                      ,                                                                                        (        1.5        )            
Then, they define the relaxed functional associated with Eqn. (1.5) as:Ē(u)=inf{limh→∞infE(uh):uh→uεL1}  (1.6)
They show the existence of an optimal solution and the coincidence between E and the associated relaxed functional Ē on C2 functions. The authors in [10] derived the Euler-Lagrange equation associated with (1.4) in the case N=2, p>1, and proposed some explicit numerical schemes to solve the corresponding Euler Lagrangian equation. However, their resulting Euler-Lagrange equation is nonlinear fourth order, so the computational cost becomes an issue. Another approach by relaxation is reported in [11], where the authors proposed a relaxed formulation of (1.4) and the term
      ∇    u                  ∇      u          is replaced with a vector field θ.